The Division Relation: Congruence Conditions and Axiomatisability

Abstract

We examine a universal algebraic abstraction of the semigroup theoretic concept of “divides:” a divides b in an algebra A if for some n ∈ ω, there is a term t(x, y ₁,…, y _n ) involving all of the listed variables, and elements c ₁,…, c _n such that t ^A (a, c ₁,…, c _n ) = b. The first order definability of this relation is shown to be a very broad generalisation of some familiar congruence properties, such as definability of principal congruences. The algorithmic problem of deciding when a finitely generated variety has this relation definable is shown to be equivalent to an open problem concerning flat algebras. We also use the relation as a framework for establishing some results concerning the finite axiomatisability of finitely generated varieties.

Key Words:

• Definable principal congruences
• Finite axiomatisability
• Flat algebras
• Semigroups and universal algebra

2000 Mathematics Subject Classification:

• 08A30
• 08C15
• 20M07

ACKNOWLEDGMENT

Marcel Jackson was supported by ARC Discovery Project Grant DP0342459.

Notes

¹Strictly, z ₀ xz ₁ should be either z ₀(xz ₁) or (z ₀ x)z ₁, but we can work with semigroup words instead of terms.

²Note also, that the definability of I _a and J _a are obviously equivalent to the definability of the corresponding congruences. For example, if θ _a denotes the Rees congruence corresponding to I _a and Φ(x, y) defines principal ideals, then the sentence Φ′(x, y, z): = (x ≈ y)∨Φ(z, x) & Φ(z, y) defines θ in the sense that (c, d) ∈ θ _a if and only if A⊨Φ′(c, d, a).

³In this article we do not consider the one element algebra to be subdirectly irreducible.

⁴The statement of Corollary 2.8 of Demlová and Koubek [14] requires the phrase “where P is in normal form” to be added for it to read correctly.

Communicated by V. Gould.